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Communications in Algebra Volume 36, 2008 - Issue 2
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Original Articles

Classification of Rings with Genus One Zero-Divisor Graphs

Cameron Wickham Department of Mathematics, Missouri State University, Springfield, Missouri, USACorrespondence[email protected]
Pages 325-345 | Received 12 Oct 2006, Published online: 07 Apr 2008

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Nadeem ur Rehman, Mohd Nazim & K. Selvakumar. (2022) On the planarity, genus and crosscap of new extension of zero-divisor graph of commutative rings. AKCE International Journal of Graphs and Combinatorics 19:1, pages 61-68.
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L. Hamidian Jahromi & A. Abbasi. (2020) On the genus of some total graphs. AKCE International Journal of Graphs and Combinatorics 17:1, pages 560-570.
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K. Selvakumar & M. Subajini. (2018) Commutative rings with genus two annihilator graphs. Communications in Algebra 46:1, pages 28-37.
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Chanlemki Lanong & Sanghita Dutta. (2017) Some results on graphs associated with vector spaces. Journal of Information and Optimization Sciences 38:8, pages 1357-1368.
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K. Selvakumar & V. Ramanathan. (2017) Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs. Communications in Algebra 45:1, pages 275-284.
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Ashish Kumar Das & Deiborlang Nongsiang. (2015) On the Genus of the Nilpotent Graphs of Finite Groups. Communications in Algebra 43:12, pages 5282-5290.
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F. Aliniaeifard & M. Behboodi. (2013) Commutative Rings Whose Zero-Divisor Graphs Have Positive Genus. Communications in Algebra 41:10, pages 3629-3634.
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Nathan Bloomfield. (2013) The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5 . Communications in Algebra 41:2, pages 765-775.
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T. Tamizh Chelvam & T. Asir. (2013) On the Genus of the Total Graph of a Commutative Ring. Communications in Algebra 41:1, pages 142-153.
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Nathan Bloomfield & Cameron Wickham. (2010) Local Rings with Genus Two Zero Divisor Graph. Communications in Algebra 38:8, pages 2965-2980.
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Articles from other publishers (31)

K. Selvakumar & N. Anusha. (2023) On the ideal-based zero-divisor graph of commutative rings. Discrete Mathematics, Algorithms and Applications 16:01.
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Thangaraj Asir, Karuppiah Mano & Turki Alsuraiheed. (2023) Classification of Genus Three Zero-Divisor Graphs. Symmetry 15:12, pages 2167.
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Praveen Mathil & Jitender Kumar. (2023) Characterization of rings with genus two prime ideal sum graphs. Asian-European Journal of Mathematics 16:11.
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Nadeem ur Rehman, Mohd Nazim & K. Selvakumar. (2022) On the genus of extended zero-divisor graph of commutative rings. Rendiconti del Circolo Matematico di Palermo Series 2 72:7, pages 3541-3550.
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K. Selvakumar, V. Ramanathan & C. Selvaraj. (2022) On the genus of dot product graph of a commutative ring. Indian Journal of Pure and Applied Mathematics 54:2, pages 558-567.
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E. Jesili, K. Selvakumar & T. Tamizh Chelvam. (2022) On the genus of reduced cozero-divisor graph of commutative rings. Soft Computing 27:2, pages 657-666.
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Ayman Badawi & Yasmine El-Ashi. 2022. Algebra and Related Topics with Applications. Algebra and Related Topics with Applications 401 412 .
Abdulaziz M. Alanazi, Mohd Nazim & Nadeem Ur Rehman. (2021) Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings. Journal of Mathematics 2021, pages 1-7.
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Abdulaziz M. Alanazi, Mohd Nazim & Nadeem Ur Rehman. (2021) Classification of Rings with Toroidal and Projective Coannihilator Graph. Journal of Mathematics 2021, pages 1-7.
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T. Asir, K. Mano & M. Subathra. 2021. Rings, Monoids and Module Theory. Rings, Monoids and Module Theory 261 285 .
G. Kalaimurugan, P. Vignesh & T. Tamizh Chelvam. (2019) On zero-divisor graphs of commutative rings without identity. Journal of Algebra and Its Applications 19:12, pages 2050226.
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G. Kalaimurugan, P. Vignesh & T. Tamizh Chelvam. (2020) Toroidal zero-divisor graphs of decomposable commutative rings without identity. Boletín de la Sociedad Matemática Mexicana 26:3, pages 807-829.
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T. Asir & K. Mano. (2019) Classification of non-local rings with genus two zero-divisor graphs. Soft Computing 24:1, pages 237-245.
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T. Asir & K. Mano. (2019) Classification of rings with crosscap two class of graphs. Discrete Applied Mathematics 265, pages 13-21.
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K. Selvakumar, M. Subajini & M. J. Nikmehr. (2018) Finite commutative ring with genus two essential graph. Journal of Algebra and Its Applications 17:07, pages 1850121.
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Huadong Su & Yiqiang Zhou. (2018) Finite commutative rings whose unitary Cayley graphs have positive genus. Journal of Commutative Algebra 10:2.
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K. Selvakumar & V. Ramanathan. (2016) Classification of non-local rings with projective 3-zero-divisor hypergraph. Ricerche di Matematica 66:2, pages 457-468.
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Krishnan Selvakumar & Manoharan Subajini. (2016) Classification of rings with toroidal Jacobson graph. Czechoslovak Mathematical Journal 66:2, pages 307-316.
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Huadong Su, Kenta Noguchi & Yiqiang Zhou. (2014) Finite commutative rings with higher genus unit graphs. Journal of Algebra and Its Applications 14:01, pages 1550002.
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R. Kala & S. Kavitha. (2014) Nilpotent graphs of genus one. Discrete Mathematics, Algorithms and Applications 06:03, pages 1450037.
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Huadong Su & Pailing Li. (2014) On the Genus of the Zero-Divisor Graph of . International Journal of Combinatorics 2014, pages 1-5.
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Ashish Das, H. R. Maimani, Mohammad Pournaki & Siamak Yassemi. (2014) Nonplanarity of unit graphs and classification of the toroidal ones. Pacific Journal of Mathematics 268:2, pages 371-387.
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Zoran S. Pucanović & Zoran Z. Petrović. (2013) Toroidality of Intersection Graphs of Ideals of Commutative Rings. Graphs and Combinatorics 30:3, pages 707-716.
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T. Asir & T. Tamizh Chelvam. (2013) On the genus of generalized unit and unitary Cayley graphs of a commutative ring. Acta Mathematica Hungarica 142:2, pages 444-458.
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Kazem Khashyarmanesh & Mahdi Reza Khorsandi. (2013) Projective total graphs of commutative rings. Rocky Mountain Journal of Mathematics 43:4.
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T. TAMIZH CHELVAM & T. ASIR. (2013) THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-I. Journal of Algebra and Its Applications 12:04, pages 1250198.
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Hamid Reza Maimani, Cameron Wickham & Siamak Yassemi. (2012) Rings whose total graphs have genus at most one. Rocky Mountain Journal of Mathematics 42:5.
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F. ALINIAEIFARD & M. BEHBOODI. (2012) RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS. Journal of Algebra and Its Applications 11:03, pages 1250049.
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David F. Anderson, Michael C. Axtell & Joe A. SticklesJr.Jr.. 2011. Commutative Algebra. Commutative Algebra 23 45 .
Hung-Jen Chiang-Hsieh, Pei-Feng Lee & Hsin-Ju Wang. (2010) The embedding of line graphs associated to the zero-divisor graphs of commutative rings. Israel Journal of Mathematics 180:1, pages 193-222.
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Cameron Wickham. (2009) Rings whose zero-divisor graphs have positive genus. Journal of Algebra 321:2, pages 377-383.
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